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Mirrors > Home > ILE Home > Th. List > lssvscl | GIF version |
Description: Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lssvscl.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lssvscl.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lssvscl.b | ⊢ 𝐵 = (Base‘𝐹) |
lssvscl.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
lssvscl | ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → (𝑋 · 𝑌) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 527 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → 𝑊 ∈ LMod) | |
2 | simprl 529 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → 𝑋 ∈ 𝐵) | |
3 | simplr 528 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → 𝑈 ∈ 𝑆) | |
4 | simprr 531 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → 𝑌 ∈ 𝑈) | |
5 | eqid 2189 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
6 | lssvscl.s | . . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | 5, 6 | lsselg 13677 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ (Base‘𝑊)) |
8 | 1, 3, 4, 7 | syl3anc 1249 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → 𝑌 ∈ (Base‘𝑊)) |
9 | lssvscl.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
10 | lssvscl.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
11 | lssvscl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐹) | |
12 | 5, 9, 10, 11 | lmodvscl 13621 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ (Base‘𝑊)) → (𝑋 · 𝑌) ∈ (Base‘𝑊)) |
13 | 1, 2, 8, 12 | syl3anc 1249 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → (𝑋 · 𝑌) ∈ (Base‘𝑊)) |
14 | eqid 2189 | . . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
15 | eqid 2189 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
16 | 5, 14, 15 | lmod0vrid 13635 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑋 · 𝑌) ∈ (Base‘𝑊)) → ((𝑋 · 𝑌)(+g‘𝑊)(0g‘𝑊)) = (𝑋 · 𝑌)) |
17 | 1, 13, 16 | syl2anc 411 | . 2 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → ((𝑋 · 𝑌)(+g‘𝑊)(0g‘𝑊)) = (𝑋 · 𝑌)) |
18 | 15, 6 | lss0cl 13685 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (0g‘𝑊) ∈ 𝑈) |
19 | 18 | adantr 276 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → (0g‘𝑊) ∈ 𝑈) |
20 | 9, 11, 14, 10, 6 | lssclg 13680 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈 ∧ (0g‘𝑊) ∈ 𝑈)) → ((𝑋 · 𝑌)(+g‘𝑊)(0g‘𝑊)) ∈ 𝑈) |
21 | 1, 3, 2, 4, 19, 20 | syl113anc 1261 | . 2 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → ((𝑋 · 𝑌)(+g‘𝑊)(0g‘𝑊)) ∈ 𝑈) |
22 | 17, 21 | eqeltrrd 2267 | 1 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → (𝑋 · 𝑌) ∈ 𝑈) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 ‘cfv 5235 (class class class)co 5896 Basecbs 12512 +gcplusg 12589 Scalarcsca 12592 ·𝑠 cvsca 12593 0gc0g 12761 LModclmod 13603 LSubSpclss 13668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-i2m1 7946 ax-0lt1 7947 ax-0id 7949 ax-rnegex 7950 ax-pre-ltirr 7953 ax-pre-ltadd 7957 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-pnf 8024 df-mnf 8025 df-ltxr 8027 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-5 9011 df-6 9012 df-ndx 12515 df-slot 12516 df-base 12518 df-sets 12519 df-plusg 12602 df-mulr 12603 df-sca 12605 df-vsca 12606 df-0g 12763 df-mgm 12832 df-sgrp 12865 df-mnd 12878 df-grp 12948 df-minusg 12949 df-sbg 12950 df-mgp 13275 df-ur 13314 df-ring 13352 df-lmod 13605 df-lssm 13669 |
This theorem is referenced by: lssvnegcl 13692 islss3 13695 islss4 13698 lspsneli 13731 lspsn 13732 |
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